# Powers of ten and physical units

Authored explanation; editorial review pending.

Microscopic measurements in physics involve scales far outside everyday experience. Writing numbers as a coefficient between 1 and 10 multiplied by a power of ten keeps significant digits clear and avoids counting long strings of zeros.

Units attach physical dimensions to numbers. A distance of 1 micrometre (10⁻⁶ m) squared is 10⁻¹² m², not 10⁻⁶ m². Calculations must keep units alongside numbers through every algebraic step.

## Worked example

1. 1 micrometre (1 μm) equals 10⁻⁶ metres (0.000001 m).
2. A typical Brownian displacement of 6 μm after one minute is 6 × 10⁻⁶ m.
3. Squaring this displacement gives (6 × 10⁻⁶ m)² = 36 × 10⁻¹² m² = 3.6 × 10⁻¹¹ m².

Every physical number must carry its units, and powers of ten apply to both coefficient and unit.